R. COLLINS21

compiled for the Survey of Occupational Injuries and

Illnesses (SOII) using methods described in Chapter 9

of the BLS Handbook of Methods [4].

It should be noted that these annual summaries are the

result of statistical analyses of raw data collected by BLS

from representative businesses within the industries

mentioned. Access to the underlying raw data is severely

restricted. Moreover, the raw data collected by BLS is

only a sample collected to represent all businesses/estab-

lishments in the United States. Therefore, the data sum-

maries provided by BLS are statistical summaries of

samples collected in a manner so as to be representative

of all businesses/establishments in the United States.

Without access to the underlying raw data, we cannot

know the shape of the data’s distribution or the distribu-

tion of data for all businesses/establishments in the

United States. Without knowledge of the shape of the

underlying data, we cannot know whether to use para-

metric or nonparametric statistical methods to analyze

the data in these summaries. Fortunately, the Central

Limit Theorem allows to assume that the sample means

and confidence limits obtained from summaries of the

underlying raw data are normally distributed because the

sample size is so very large. Had the underlying raw data

been available for analyses, the work presented in this

article would have benefited greatly. However, the

amount and variety of summary data available from the

underlying raw data is sufficient to allow us to perform

useful statistical analyses.

2.2. Data Management

Trends during 1992-2009 were analyzed to determine the

annualized and overall ratios of recordable incidents and

lost workday cases to fatal occupational injuries. This

data was used to determine the probability of experienc-

ing a fatal occupational injury per incident resulting in an

OSHA recordable incident for the time interval analyzed.

If we assume that accident prevention programs do not

change substantially over the next few years, these

probabilities derived from past incidents can be used to

predict future trends. With this assumption, probabili-

ties derived from these past incidents are used with gene-

rally accepted statistical methods to estimate the future

likelihood of fatal accident occurrence.

2.3. Data Analysis

The statistical methods chosen for this analysis are those

derived from Binomial distributions and Poisson distri-

butions. Binomial distribution methods are used because

they represent 1) a series of n independent events or tri-

als, 2) each of which can have only two possible out-

comes. In this case, a chronological series of OSHA re-

cordable incidents is the series of n independent events

or trials used. “Two events are statistically independent if

the probability of their occurring jointly equals the pro-

duct of their respective probabilities (i.e., Pr(A∩B) =

Pr(A) X Pr(B)).” [5] Each independent event is an inci-

dent resulting in an OSHA recordable incident that can

either result in a fatal or non-fatal injury. As can be seen,

a Binomial distribution is suitable for this analysis.

For recordable incidents alone, there are only two pos-

sibilities going forward. These are:

1) The recordable incident will result in a fatality; or

2) The recordable incident will not result in a fatality.

These are mutually exclusive and independent events,

otherwise known as a Bernoulli Variable. A toss of a

coin is a simpler example of a Bernoulli Variable. In

each case, the event can take on only 1 of 2 possible

values. Each toss of the coin or, in this case, recordable

incident represents a single Bernoulli Trial producing a

single result. When looking at a string of successive

Bernoulli Trials, the appropriate statistical method to use

in predicting potential future events of this type is a Bi-

nomial distribution. If we wanted to calculate the pro-

bability of having n successive trials before encountering

z specified outcomes, we would use the following gene-

ralized formula:

!

Probability 1

!!

nz

z

npp

zn z

where: n = Total number of consecutive trials or, in this

case, recordable incidents without a single fatality;

z = Total number of successes (or failures) or, in this

case, one or more fatalities;

p = Probability of a single success or failure (i.e., the

probability of one or more fatalities given n consecutive

recordable incidents).

Unfortunately, analysis using a Binomial distribution

alone gets us only part of the distance to a true temporal

probability relationship. In order to establish a temporal

probability relationship, a Poisson distribution must be

used. Poisson distributions are usually associated with

rare events. In this case, the rare event is a fatal occupa-

tional incident. Fortunately, Poisson distributions can be

used to approximate a Binomial distribution, under cer-

tain conditions. Generally accepted statistics theory al-

lows the use of a Poisson distribution to approximate a

Binomial distribution when the value of n (i.e., total

number of consecutive trials) is large and the value of p

(i.e., the probability of a single success or failure) is

small. In this case, the value of p is certainly small (i.e., p

= 1/987 = 1.01 × 10−3 ± 8.14%) and the value of n in-

creases with time. This means that the approximation of

a Binomial distribution using a Poisson distribution im-

Copyright © 2011 SciRes. OJSST

R. COLLINS

Copyright © 2011 SciRes. OJSST

22

calculate. proves with time.

In this case, we will use the Binomial distribution

formula presented above to determine the probability of

having n successive OSHA recordable incidents without

a single fatality. In order to determine this probability,

we must first determine the probability of having n suc-

cessive OSHA recordable incidents without a single

success or failure (i.e., zero fatalities) using the single

event probability provided above (i.e., p = 1/987 = 1.01 ×

10−3 ± 8.14%). This simplifies the formula as follows:

Probability using a Poisson distribution can be calcu-

lated using the following formula:

e

Probability !

k

k

where: µ = Mean of the Poisson distribution = λt;

Λ = Number of events per unit time or number of fa-

talities per unit time;

t = Selected time interval;

k = Total number of successes (or failures) or, in this

case, one or more fatalities.

00

3

zero fatalities

Probability1.01100.999 n

Microsoft Excel and Stata/IC software (version 11,

StataCorp LP, College Station, Texas, USA) were used

for all of the statistical data analyses presented in this

paper. An alpha of p < 0.05 was selected as the level of

significance.

As this is the probability of having zero fatalities in n

successive OSHA recordable incidents, the probability of

having one (1) or more fatalities in n successive OSHA

recordable incidents can be determined using the fol-

lowing formula:

one (1) or more fatalities

00

3

zero fatalities

Probability

1Probability11.01100.999 n

3. Results

BLS data for all private industry has been extracted and

summarized in Table 1 from on-line summary tables.

When this data is normalized to the number of fatalities,

Figure 1 was created to show the ratio of recordable

incidents and lost workday cases to fatalities along with

the average values over the 18-year period shown. The

means and 95% confidence intervals for these ratios are

shown in Table 2. Over the 18-year interval studied,

there were an average of 987 recordable incidents and

476 lost workday cases for each fatality. If we assume

these ratios to be predictive as well as historical, we can

use them to determine the probability of future private

industry fatalities. The 95% confidence intervals (as

percents of the mean, shown in Table 2) can be consi-

dered to be error associated with the probabilities we

Collectively, these formulas produce the graph shown

in Figure 2. The x axis in Figure 2 represents the num-

ber of successive OSHA recordable incidents. In this

graph, the solid line represents the probability of having

n successive OSHA recordable incidents without having

a single fatality (i.e., zero fatalities). The dashed line is

the statistical complement to this (i.e., 1 – Probabilityzero

fatalities) showing the probability of having 1 or more fa-

talities in n successive OSHA recordable incidents. The

lighter lines above and below both of these lines repre-

sent the upper and lower bounds, given the error pro-

duced at the 95% confidence level (i.e.,

= 0.05). As

can be seen, the probability of having n successive

OSHA recordable incidents without a single fatality

Table 2. Ratios for all private industry and select industry groups.

95% Confidence Limits

Mean Lower Limit Upper Limit % Error

All Private Industry:

Ratio of OSHA Recordable Inc i dents to Fatalities 987 906 1,067 ± 8.14

Ratio of OSHA Lost Workday Cases to Fatalities 476 451 502 ± 5.27

NAICS Code 238 (Specialty Trade Contractors Industry Group):

Ratio of OSHA Recordable Incidents to Fatalities 442 407 477 ± 7.83

Ratio of OSHA Lost Workday Cases to Fatalities 221 209 232 ± 5.11

NAICS Code 211 (Oil and Gas Extraction Industry Group):

Ratio of OSHA Recordable Incidents to Fatalities 167 142 191 ± 14.4

Ratio of OSHA Lost Workday Cases to Fatalities 93 76 109 ± 17.4

R. COLLINS23

Figure 2. Fatality probabilities per consecutive recordable incident. Upper and lower confience limits (i.e., % error) are

shown as narrower lines above and below.

drops significantly while the probability of having one or

more fatalities in n successive OSHA recordable inci-

dents climbs exponentially approaching 1 (or 100%)

asymptotically.

In order to use a Poisson distribution to approximate

the Binomial distribution results, we have all of these

values in the formula provided above except for the

value of λ (i.e., the number of events per unit time). We

do have the value of p (i.e., the probability of a single

success or failure per recordable incident), however. We

can use the value of p to get the value of λ if we know

the temporal frequency of recordable incidents. The

formula for determining the value of λ from the value of

p, given the temporal frequency of recordable incidents

(f), is:

pf

where: λ = Number of events per unit time

p = Probability of a single success or failure per re-

cordable incident

f = Temporal frequency of recordable incidents per

unit of time

The astute reader will recognize that the temporal fre-

quency of recordable incidents is directly related to size

of the business (i.e., the number of hours worked each

year). It may also be a function of the less quantifiable

variables of 1) intrinsic risk of the business; and 2) qua-

lity of the accident prevention program.

For the purposes of this example, let’s assume that the

value of f is 100 recordable incidents per year. Substi-

tuting into the formula above, we get a value for λ equal

to:

3

1

1.01 10Fatalities100Recordable Incidents

Recordable IncidentYear

Fatalities

1.0110Year

Substituting this value for λ into the probability func-

tion above yields the following results:

1

1.01 10fatalitiesYear1

Probability

e1.0110fatalities Year

!

k

tt

k

Figure 3(a) shows the results of this equation graphi-

cally. In this case, the x axis is time in years. The solid

line represents the probability of having zero (0) fatali-

ties (i.e. Probabilityzero fatalities) in the time intervals indi-

cated in years. The dashed line is the statistical comple-

ment to this (i.e., 1 – Probabilityzero fatalities) showing the

probability of having 1 or more fatalities (i.e., Probabili-

tyone (1) or more fatalities) in the time intervals indicated in

years. The lighter lines above and below both of these

lines represent the upper and lower bounds, given the

error produced at the 95% confidence level (i.e.,

=

0.05). As can be seen, the probability of having zero (0)

Copyright © 2011 SciRes. OJSST

R. COLLINS

24

fatalities drops significantly while the probability of

having one or more fatalities in the time intervals indi-

cated in years climbs exponentially approaching 1 (or

100%) asymptotically. For this example of 100 record-

able incidents per year, the probability of having one or

more fatalities exceeds 50% after only 7 years.

If we were to use a different example, say 10 record-

able incidents per year, the graph (Figure 3(b)) would

appear the same but the x axis would reach further into

the future. In this example, when the x axis is expanded

to 500 years the graph is very similar in appearance. For

10 recordable incidents per year, the probability of hav-

ing one or more fatalities exceeds 50% after only 69

years.

4. Discussion

It should be noted that the ratios of 477 lost workday

cases per fatality and 987 recordable incidents per fatal-

ity (i.e., 1:477:987) were derived for all private industry.

In reality, private industry within the United States is

made up of dozens of business types and thousands of

(a)

(b)

Figure 3. Fatal probability per year for all Private Industry. Solid line indicates the probability of zero (0) fatalities per year

while the dashed line indicates the probability of one (1) or more fatalities per year. Lighter shaded lines above and below

represent the upper and lower confidence limits (i.e., % error). Figure 3(a) assumes a rate of 100 recordable incidents per

ear. Figure 3(b) assumes a rate of 10 recordable incidents per year. y

Copyright © 2011 SciRes. OJSST

R. COLLINS

Copyright © 2011 SciRes. OJSST

25

individual businesses. It is unlikely that all business

types and individual businesses would have the same risk

profile. We can see this by looking at two specific exam-

ples. In the most recent year for which records are avail-

able (i.e., 2009), Specialty Trade Contractors (North

American Industrial Classification System or NAICS

Code 238) had the highest total number of fatalities (i.e.,

477) while Oil and Gas Exploration firms (NAICS Code

211) had among the lowest total number of fatalities (i.e.,

17). When we apply the same methodology used above

to characterize risk within private industry, we get the

risk profiles shown in Table 2. From these results, it can

be seen that the risk profile for private industry (i.e.,

1:476:987) is significantly different from the risk profiles

for Specialty Trade Contractors (i.e., 1:221:442) and Oil

and Gas Exploration (i.e., 1:93:167). These risk profiles

have nothing to do with the size of the business (i.e., the

number of hours worked each year) and everything to do

with those less quantifiable variables of 1) the intrinsic

risk of their businesses; and/or 2) the quality of their ac-

cident prevention programs.

In any event, different risk profiles mean different

probabilities for a single success or failure (i.e., p) that

can be applied as above. The resulting probability graphs

(Figure 4(b) for Specialty Trade Contractors and Figure

4(c) for Oil and Gas Exploration), assuming 100 record-

able incidents per year, appear very similar to that for all

private industry (i.e., Figure 4(a)) with only a shift to the

right or left on the x axis. However, as can be seen, the

probability of having one or more fatalities exceeds 50%

after 3 years for Specialty Trade Contractors (NAICS

Code 238) and 1 year for Oil and Gas Extraction

(NAICS Code 211), assuming an annual rate of 100 re-

cordable incidents per year. This would seem to indicate

that working in the Oil and Gas Extraction industry

group (NAICS Code 211) is more dangerous (at least

from the likelihood of death following an incident) than

it is in the Specialty Trade Contractors industry group

(NAICS Code 238) and working in either industry group

appears to be more dangerous than working in private

industry as a whole. This is an especially interesting re-

sult, given that two industry groups with diametrically

opposite fatality totals during 2009 (i.e., 17 and 477)

were chosen. This may be explained by the fact that,

based upon hours worked, the Oil and Gas Extraction

industry group (NAICS Code 211) is 1/10th of the size of

the Specialty Trade Contractors industry group (NAICS

Code 238) and this group is 1/100th of the size of all pri-

vate industry.

This same type of refined analysis can be applied to an

individual business/establishment if there is sufficient

historical data on which to base a potential future prob-

ability. Figure 5 shows how this might be applied to a

single business/establishment. The x axis in this figure

represents years while the y axis represents probability.

The solid line descending from left to right represents the

probability of experiencing zero (0) fatal incidents for

the general industry type corresponding to a specific

business/establishment while the dashed line descending

from left to right represents the same probability for a

specific business/establishment. The solid line ascending

from left to right represents the probability of one (1) or

more fatal incidents for the general industry type corre-

sponding to a specific business/establishment while the

dashed line descending from left to right represents the

same probability for a specific business/establishment.

The vertical dashed line represents the current position in

time on the graph for this specific business/establishment.

Two (2) potentially valuable lessons that can be learned

from this figure are:

1) The dashed lines representing a specific business/

establishment are to the left of the solid lines represent-

ing the general industry type to which the specific busi-

ness/establishment belongs. This can be interpreted as

meaning that it is more dangerous to work in this specific

business/establishment than it is for the group to which

this business/establishment belongs.

2) At the specific point in time indicated by the verti-

cal dashed line, the probability that the next OSHA re-

cordable incident will result in a fatal accident is 35.6%

for the specific business/establishment while it would

only be 19.9% for the group to which this business/es-

tablishment belongs.

It should be noted, however, that an individual busi-

ness/establishment will be a subset of the industry group

to which it belongs. Moreover, both the Oil and Gas Ex-

traction industry group (NAICS Code 211) and the Spe-

cialty Trade Contractors industry group (NAICS Code

238) are subsets of the larger private industry group.

While restricting this analysis to a specific industry

group or business/establishment might produce more

useful results, the confidence interval size (i.e., error)

must of necessity increase as well making the results

potentially less reliable. Statistically speaking, confi-

dence intervals tend to increase as the sample size de-

creases. In practice, however, two (2) different busi-

nesses within the same industry group may have widely

different approaches to accident prevention. A business

with limited efforts to prevent accidents, sometimes

known as a poor safety culture, may find their accident

prediction curve (i.e., the Poisson distribution demon-

strated above) shifted to the left compared to the industry

group. Whereas, a business with strong efforts to prevent

accidents, sometimes known as an excellent safety cul-

ture, may find their accident prediction curve shifted to

the right. In this example, smller sample sizes are more a

R. COLLINS

26

(a)

(b)

(c)

Figure 4. Fatal probability per year, assuming a rate of 100 recordable incidents per year. Solid lines indicate the probability

of zero (0) fatalities per year while the dashed lines indicate the probability of one (1) or more fatalities per year. Lighter

shaded lines above and below represent the upper and lower confidence limits (i.e., % error). Figure 4(a) is for all Private

Industry. Figure 4(b) is for the Specialty Trade Contractors industry group (NAICS Code 238). Figure 4(c) is for the Oil and

Gas Extraction industry group (NAICS Code 211).

Copyright © 2011 SciRes. OJSST

R. COLLINS27

Figure 5. Fatal probability per year for a single business/establishment.

susceptible to variations in the quality of accident pre-

vention policies, programs and procedures.

Readers of Heinrich’s original work will note that the

three (3) levels of his original triangle consisted of 1)

Major Injuries; 2) Minor Injuries; and 3) No-Injury Ac-

cidents. The definitions provided for those original three

(3) categories were:

1) Major Injuries: “Any case that is reported to insu-

rance carriers or to the state compensation commis-

sioner” [1]. In today’s vernacular, these would be OSHA

recordable cases.

2) Minor Injuries: “A scratch, bruise, or laceration

such as is commonly termed a first-aid case” [1].

3) No-Injury Accidents: “An unplanned event involve-

ing the movement of a person or an object, ray, or sub-

stance (slip, fall, flying object, inhalation, etc.), having

the probability of causing personal injury or property

damage.” [1].

From this we can see, that the three (3) levels of the

New Incident Pyramid described earlier in this article

(i.e., fatalities, lost workday cases and OSHA recordable

cases) would all fit into only the top level of Heinrich’s

original triangle. The bottom two levels of Heinrich’s

original triangle would appear as two additional layers of

this New Incident Pyramid. From Heinrich’s original

triangle, we know only that these two (2) bottom levels

of the New Incident Pyramid would be located below the

OSHA recordable case level and that they would be lar-

ger than the OSHA recordable case level. Unfortunately,

there are no reliable resources to quantify these addi-

tional levels. If we were to use the ratios provided in

Heinrich’s original work, the New Incident Pyramid

would appear as seen in Figure 6 for all private industry.

If data for these two (2) additional levels of the New In-

cident Pyramid were more current and reliable, we could

use them in the same type of analyses presented in this

article to predict potential future fatalities based upon the

total number of all incidents instead of only OSHA re-

cordable incidents. As there are no accurate and reliable

modern day resources from which data can be gathered

for these additional analyses, we can rely only on the

analyses presented in this article.

5. Conclusions

It should be noted that the statistical analyses presented

in this article were performed using summary tables de-

rived from raw data collected by BLS from representa-

tive businesses. The statistical analyses performed within

this article would have benefitted greatly from access to

the underlying raw data in lieu of these summary tables.

One of the fundamental underlying principals of

Heinrich’s original triangle is that fatalities cannot occur

without a foundation of less severe incidents. In other

words, increasing numbers of non-serious incidents

eventually support the existence of more serious and

fatal incidents. We can now see from the analyses pre-

sented in this paper that this also applies to incidents

Copyright © 2011 SciRes. OJSST

R. COLLINS

28

Figure 6. New incide nt pyramid for all private industry.

occurring over time. Valuable conclusions that can be

drawn from this analysis include:

1) Fatal accidents are inevitable. Each day that passes

without a fatal occupational accident only leads to an

increased likelihood that a fatal occupational accident

will occur during the next day as the probability ap-

proaches 100% asymptotically. This conclusion may be

dependent on the occurrence of less severe incidents as

well as recordable incidents, as demonstrated by the

adaption of Heinrich’s original triangle shown in Figure

6.

2) Factors potentially affecting when these fatalities

will occur include:

a) Size of the business (i.e., number of hours worked

each year);

b) Intrinsic risk of the business; and/or

c) Quality of the accident prevention program.

As only one (1) of these factors is within the direct

control of business managers, the likelihood of a poten-

tial future fatal accident can only be decreased by con-

tinually improving the quality of their accident preven-

tion programs.

Perhaps the central lesson to be learned from these

analyses is that we must always look on recordable inci-

dents (and even accidents without injuries) as portents of

fatal accidents to come. Doing little to prevent recordable

or minor accidents today will only lead to fatal incidents

sooner rather than later; if we do not respond appropri-

ately when they occur. It is only by considering safety

and accident prevention with our every thought and ac-

tion that we can hope to push these fatal accidents further

off into the future by shifting our own personal probabi-

lity distribution to the right.

6. References

[1] H. H. William, “Industrial Accident Prevention: A Scien-

tific Approach,”4th Edition, McGraw-Hill Book Com-

pany, Boston, 1959.

[2] R. L. Collins, “Heinrich and Beyond,” Process Safety

Progress, Vol. 30, No. 1, 2011, pp. 2-5.

[3] B. Rosner, “Fundamentals of Biostatistics,” Brooks/Cole,

Copyright © 2011 SciRes. OJSST

R. COLLINS29

Pacific Grove, 2010.

[4] United States, “BLS Handbook of Methods,” The Bureau,

Washington, 1982.

[5] “McGraw-Hill Dictionary of Scientific and Technical

Terms,” McGraw-Hill, New York, 2003.

Copyright © 2011 SciRes. OJSST